Properties

Label 4-975e2-1.1-c1e2-0-37
Degree $4$
Conductor $950625$
Sign $1$
Analytic cond. $60.6126$
Root an. cond. $2.79023$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 10·11-s − 4·16-s − 4·19-s − 20·29-s − 4·31-s − 18·41-s + 5·49-s − 22·61-s + 30·71-s + 22·79-s + 81-s + 22·89-s + 10·99-s − 24·101-s − 32·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1/3·9-s − 3.01·11-s − 16-s − 0.917·19-s − 3.71·29-s − 0.718·31-s − 2.81·41-s + 5/7·49-s − 2.81·61-s + 3.56·71-s + 2.47·79-s + 1/9·81-s + 2.33·89-s + 1.00·99-s − 2.38·101-s − 3.06·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(950625\)    =    \(3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(60.6126\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 950625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 113 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.541758690289089937798999864926, −9.539085148781746476495345714380, −9.070956808484618678779103838040, −8.386119843524000244102713291511, −8.097372236858610332846946511579, −7.79724243594435474886350774139, −7.35312019626074465076408138506, −6.91040422787831570441579040453, −6.41671683791534006534281028614, −5.65727426264097005904742242830, −5.52219927950668028705023352890, −4.95410475736013450010420585573, −4.78772740075501855139012978932, −3.72774110639731016572210183479, −3.55738854060454395267998035213, −2.74318712192696589720572087573, −2.08960137747437871260749079517, −1.98095357619638911834050674061, 0, 0, 1.98095357619638911834050674061, 2.08960137747437871260749079517, 2.74318712192696589720572087573, 3.55738854060454395267998035213, 3.72774110639731016572210183479, 4.78772740075501855139012978932, 4.95410475736013450010420585573, 5.52219927950668028705023352890, 5.65727426264097005904742242830, 6.41671683791534006534281028614, 6.91040422787831570441579040453, 7.35312019626074465076408138506, 7.79724243594435474886350774139, 8.097372236858610332846946511579, 8.386119843524000244102713291511, 9.070956808484618678779103838040, 9.539085148781746476495345714380, 9.541758690289089937798999864926

Graph of the $Z$-function along the critical line